The way you have phrased the question is rather ambiguous.

If you mean to ask whether the Roman roads/road network continued to be used, then the answer would be: Yes, the Byzantine Empire continued the use of Roman roads (or most of them). So did most of the 'states' succeeding the Roman Empire in the West. In fact - until the arrival of the highways - the Roman road network formed the backbone of that of the UK, much of it can still easily be traced in France, the main road between Cologne and the Channel coast can easily be traced along the majority of its length, and so on.

As long as there was no major reason to shift the road network due to (for instance) the decline (or disappearance) of population centers and such and the emergence of new ones elsewhere, or the collapse of major pieces of infrastructure (mostly bridges) there was no good reason for anyone to abandon Roman roads, and build new ones from scratch. The Romans actually often overlaid their 'new' roads over existing tracks.

If you mean to ask whether the Byzantines continued to build (and maintain) Roman roads in a way a Roman road is often depicted to be (raised center, ditches on the sides, foundation of gravel and sand, covered with slabs as a road surface), this question would be much harder to answer. The thing is, the (vast) majority of Roman roads was never constructed as such anyway; there was major variance in road surfaces, the ditches on the side were more common. Some of the roads in the rocky desert in Syria/Palestine basically consisted of little more than a track swept clear of loose rubble, with markers along the side of the road (and roadstations and such).

TL/DR Almost certainly the Byzantine Empire continued to use Roman roads. Other then when population centers shifted (something which did happen!), there was no reason at all for them to abandon Roman roads.